On the Non-linear Stability of Flux Reconstruction Schemes

Transcription

1 DOI /s TECHNICA NOTE On the Non-linear Stability of Flux econstruction Schemes A. Jameson P.E. Vincent P. Castonguay eceived: 9 December 010 / evised: 17 March 011 / Accepted: 14 April 011 Springer Science+Business Media, C 011 Abstract The flux reconstruction (F approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG methods, and all spectral difference methods (at least for a linear flux function, within a single framework. ecently a new range of linearly stable F schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH schemes. In this short note non-linear stability properties of F schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability. It is shown that linearly stable VCJH schemes (at least in their standard form may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since F schemes (in their standard form utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the F approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of F schemes implies stability is independent of solution point location. Finally, it is shown that if an exact projection is employed to construct an approximation of the flux (as opposed to a collocation projection, then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the F approach. It can be noted that in all above regards, non-linear stability properties of F schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of F schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function. Keywords High-order methods Flux reconstruction Nodal discontinuous Galerkin method Spectral difference method Non-linear stability A. Jameson P.E. Vincent ( P. Castonguay Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA pvincent@stanford.edu

2 1 Introduction In recent decades discontinuous Galerkin (DG methods, and a number of similar variants, have emerged as an attractive alternative to classical finite element and finite volume methods for high-order accurate numerical simulations on unstructured grids. ecently Huynh [1, ] proposed the flux reconstruction (F approach, which encompasses both collocation based nodal DG schemes of the type described by Hesthaven and Warburton [3], and spectral difference (SD methods (at least for a linear flux function, which were originally proposed by Kopriva and Kolias [4], and later generalized by iu, Vinokur and Wang [5]. Utilizing the F approach of Huynh [1, ],itwasprovedbyjameson[6] that(for1d linear advection a particular SD method is stable for all orders of accuracy in a broken norm of Sobolev type. ecently, this result has been extended by Vincent, Castonguay and Jameson [7], who identified a class of F schemes which are guaranteed to be linearly stable. These schemes, which are parameterized by a single scalar, will henceforth be referred to as Vincent-Castonguay-Jameson-Huynh (VCJH schemes. The identification of such schemes offers significant insight into why certain F schemes are stable, whereas others are not. Also from a practical standpoint the VCJH formulation offers a simple prescription for implementing an infinite range of efficient and linearly stable high-order methods. In this short note non-linear stability properties of F schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability. To begin, a brief overview of the one-dimensional (1D F approach is given, followed by an overview of 1D VCJH schemes. The non-linear stability of 1D VCJH schemes is then analyzed and discussed. Finally conclusions are drawn. Overview of the Flux econstruction Approach Consider solving the following 1D scalar conservation law u t + f x = 0 (.1 within an arbitrary periodic domain,wherex is a spatial coordinate, t is time, u = u(x, t is a conserved scalar quantity and f = f(u is the flux of u in the x direction. Further, consider partitioning into N distinct elements each denoted n ={x x n <x<x n+1 } such that N N = n, n =. (. Finally, having partitioned into separate elements, consider representing the exact solution u within each n by a polynomial of degree k denoted u δ n = uδ n (x, t (which is in general piecewise discontinuous between elements, and the exact flux f within each n by a polynomial of degree k +enoted fn δ = f n δ (x, t (which is piecewise continuous between elements, such that a total approximate solution u δ = u δ (x, t and a total approximate flux f δ = f δ (x, t can be defined within as N u δ = u δ n u, f N δ = f δ n f. (.3

3 From an implementation perspective, it is advantageous to transform each n to a standard element S ={r r 1} via the mapping ( x xn r = Ɣ n (x = 1, (.4 x n+1 x n which has the inverse ( 1 r x = Ɣn (r = x n + ( 1 + r x n+1. (.5 Having performed such a transformation, the evolution of u δ n within any individual n (and thus the evolution of u δ within can be determined by solving the following transformed equation within the standard element S where is a polynomial of degree k, û δ t + δ = 0, (.6 r û δ =û δ (r, t = u δ n (Ɣ n (r, t (.7 δ = δ (r, t = f n δ(ɣ n (r, t, (.8 h n is a polynomial of degree k + 1, and h n = (x n+1 x n /. The F approach to solving (.6 within the standard element S can be described in five stages. The first stage involves representing û δ in terms of a nodal basis as follows û δ = where l i are agrange polynomials defined as l i = k j=0,j i û δ i l i, (.9 i=0 ( r rj r i r j, (.10 r i (i = 0tok arek + istinct solution points within S,andû δ i =ûδ i (t (i = 0tok are values of û δ at the solution points r i. The second stage of the F approach involves constructing a degree k polynomial = (r, t, defined as the approximate transformed discontinuous flux within S. Specifically, is obtained via a collocation projection at the k + 1 solution points, and can hence be expressed as ˆ f = i=0 ˆ i l i (.11 where the coefficients i = i (t are simply values of the transformed flux at each solution point r i evaluated directly from the approximate solution. The flux is termed f

4 discontinuous since it is calculated directly from the approximate solution, which is in general piecewise discontinuous between elements. The third stage of the F approach involves evaluating the approximate solution at either end of the standard element S (i.e. at r =±1. These values, in conjunction with analogous information from adjoining elements, are then used to calculate numerical interface fluxes. In what follows the numerical interface fluxes associated with the left and right hand ends of S (and transformed appropriately for use in S will be denoted and respectively. The forth stage of the F approach involves adding a correction flux δc = δc (r, t of degree k + 1to, such that their sum equals the transformed numerical interface flux at r =±1, yet remains close to within the interior of S. To construct δc such that the above requirements are satisfied, consider first defining g = g (r and g = g (r as degree k + 1 correction functions that have oscillations of small amplitude within S (and hence approximate zero in some sense, as well as satisfying g ( = 1, g (1 = 0, (.1 g ( = 0, g (1 = 1, (.13 and g (r = g ( r. (.14 A suitable expression for δc can now be written in terms of g and g as δc = ( ˆ g + ( g, (.15 where = (,t and = (1,t. Using this expression, the degree k + 1 approximate transformed total flux δ within S can be constructed from the discontinuous and correction fluxes as follows δ = + δc = + ( ˆ g + ( g. (.16 The final stage of the F approach involves evaluating the divergence of δ at each solution point r i using the expression δ r (r i = ˆ fj j=0 dl j dr (r i + ( ˆ dg dr (r i + ( ˆ f f dg dr (r i. (.17 These values can then be used to advance û δ in time via a suitable temporal discretization of the following semi-discrete expression dû δ i = δ dt r (r i. (.18 The nature of a specific F scheme depends solely on three factors, namely the location of the solution points r i, the methodology for calculating the transformed numerical interface fluxes and, and the form of the flux correction functions g (and thus g. It was shown by Huynh [1] that a collocation based (under integrated nodal DG scheme is recovered in 1D if the corrections functions g and g are the right and left adau polynomials respectively. Also, it has been shown that SD type methods can be recovered (at least for a linear flux function if the corrections g and g are set to zero at a set of k points

5 within S (located symmetrically about the origin [1]. Several additional forms of g (and thus g have also been suggested, leading to the development of new schemes, with various stability and accuracy properties. For further details of these new schemes see the articles by Huynh [1, ]. 3 Vincent-Castonguay-Jameson-Huynh Schemes VCJH schemes [7] can be recovered if the left and right corrections functions g and g respectively are defined as and where g = (k g = 1 [ k [ k + ( ηk k + k η k ( ηk k + k η k ], (3.1 ], (3. η k = c(k + 1(a kk!, a k = (k! k (k!, (3.3 k is a egendre polynomial of degree k,andc is a free scalar parameter that must lie within the range <c<. (3.4 (k + 1(a k k! Such correction functions satisfy and g û δ r dr c g û δ r dr c ( kû δ ( kû δ ( d k+1 g = 0, (3.5 ( d k+1 g = 0, (3.6 within the standard element S for any transformed solution û δ, and ensure that the resulting VCJH scheme will be linearly stable in the broken Sobolev type norm u δ k,,definedas u δ k, = [ N n=1 xn+1 x n (u δ n + c (J n k ( k u δ n dx x k ] 1/. (3.7 It can be noted that several existing methods are encompassed by the new class of VCJH schemes. In particular if c = 0 then a collocation based nodal DG scheme is recovered [7]. Alternatively, if k c = (k + 1(k + 1(a k k!, (3.8 an SD method is recovered (at least for a linear flux function [7]. It is in fact the only SD type scheme that can be recovered from the range of VCJH schemes. Further, it is identical

6 to the SD scheme that Jameson [6] proved to be linearly stable, which is the same as the only SD scheme that Huynh found to be devoid of weak instabilities [1]. Finally, if c = (k + 1 (k + 1k(a k k!, (3.9 then a so called g F method is recovered [7], which was originally identified by Huynh [1] to be particularly stable. 4 Non-linear Stability of Vincent-Castonguay-Jameson-Huynh Schemes To gain insight into the non-linear stability of VCJH schemes consider substituting (.16 into (.6, to obtain û δ t = r ( f dg dr ( ˆ f dg dr. (4.1 On multiplying (4.1 by the approximate transformed solution û δ and integrating over S one obtains and thus d dt û δ ûδ dr = û δ ˆ t r (û δ dr = ( + ( f dr ( ûδ r dr + ( ˆ û δ dg dr dr û δ dg dr, (4. dr û δ g û δ g r dr r dr + ( ˆ ûδ ûδ, (4.3 where û δ =ûδ (,t and û δ =ûδ (1,t. On differentiating (4.1 k times (in space one obtains t = k+1 +1 ( dk+1 g ( where it can be noted that since ˆ f is a polynomial of degree k f dk+1 g, (4.4 drk+1 k+1 = 0, ( and thus t = ( dk+1 g ( f dk+1 g. (4.6 drk+1

7 On multiplying (4.6 bythekth derivative of the approximate transformed solution û δ and integrating over S one obtains ( kû δ t dr = ( ( ( dl k+1 g ( d k+1 g dr dr, (4.7 and thus since û δ is a polynomial of degree k, andg and g are polynomials of degree k + 1, one obtains dt ( kû δ dr = ( ( f ( kû δ ( kû δ ( d k+1 g ( d k+1 g. (4.8 On multiplying (4.8 by the scalar quantity c (which lies in the range defined by (3.4 and summing with (4.3, one obtains d 1 dt (û δ + c dr = + ( + ( ûδ r dr + ( ˆ [ û δ g ûδ ûδ [ û δ g r dr c ( d k+1 ] r dr c g ( d k+1 g ], (4.9 which for VCJH type schemes, due to (3.5 and(3.6, can be written as 1 dt (û δ + c dr = ûδ r dr + ûδ ûδ. (4.10 To proceed, consider writing (4.10as dt (û δ + c dr = ûδ r dr + ˆ ûδ ûδ +ˆɛ, (4.11 where f ˆ = f(r,t= ˆ f(uδ n (Ɣ n (r, t (4.1 J n is the transformed (true flux function, and ˆɛ = ( f ˆ ûδ dr (4.13 r

8 is a transformed error term (to be discussed in more detail shortly. On transforming (4.10 back to the physical space element n, and summing over all elements within the periodic domain, one obtains where dt uδ k, N = [ xn+1 x n f(u δ n uδ n x ] dx + fn uδ n (x n fn+1 uδ n (x n+1 + ɛ n, (4.14 fn = J nf ˆ, f n+1 = J nf ˆ, (4.15 are numerical interface fluxes in physical space evaluated at x n and x n+1 respectively, and are error terms in physical space within each n. If one now defines G = G(u such that ɛ n = J n ˆɛ (4.16 then (4.14 can be written as G = f, (4.17 u dt uδ k, N = [ xn+1 x n ] G u (uδ n uδ n dx + fn x uδ n (x n fn+1 uδ n (x n+1 + ɛ n, (4.18 and thus dt uδ k, N [ = G(u δ n (x n+1 G(u δ n (x n + fn uδ n (x n fn+1 uδ n (x ] n+1 + ɛ n, (4.19 which can be cast (partially in terms of a summation over interfaces within the periodic domain as dt uδ k, N [ = f n (uδ + (x n u δ (x n G(u δ + (x n + G(u δ (x n ] N + ɛ n, (4.0 where u δ + (x n = u δ n (x n and (to account for the periodicity of the domain u δ (x n = { u δ N (x N, n = 0, u δ n (x n, n 0. (4.1

9 Finally, using the mean value theorem, (4.0 can be written as dt uδ k, N [ = fn (uδ + (x n u δ (x n G ] u (ηδ n (uδ + (x n u δ (x n for some η δ n between uδ (x n and u δ + (x n, thus N dt uδ k, = (f ɛ n, (4. N + N n f(ηδ n (uδ + (x n u δ (x n + ɛ n. (4.3 If each interface flux is now considered to be an E-flux [8], then all interface contributions will be negative (following the definition of an E-flux, and hence (4.3 can be written as N dt uδ k, = + ɛ n, (4.4 where 0. For energy stability in the norm u δ k,, it is therefore required that the sum of ɛ n is less than or equal to zero. 5 The Error Terms ɛ n The nature of the error terms ɛ n (which clearly determine whether the scheme is stable can be understood by analyzing the transformed error ˆɛ within S.Sinceû δ is a polynomial of degree k, it has a spatial derivative of degree k 1, which can be expanded as û δ k r = [ (i + 1 i=0 û δ ] r idr i, (5.1 where i are egendre polynomials of degree i. On substituting (5.1into(4.13 one obtains and hence where k ˆɛ = i=0 [ ( f ˆ (i + 1 k ˆɛ = i=0 ˆɛ i = û δ ] r idr i dr, (5. [ (i û δ ] ˆɛ i r idr, (5.3 i dr ˆ i dr. (5.4 Neither the sign nor magnitude of the integral term in (5.3 can be guaranteed (since it depends on the transformed approximate solution û δ. Therefore, in order to in general minimize ˆɛ and thus ɛ n, one should ensure the magnitude of all ˆɛ i are as small as possible.

10 If the flux function is linear then will be a polynomial of degree k. Hence it will be represented exactly by (formed by a collocation projection at the k + 1 solution points. It is therefore clear that ˆɛ, and hence ɛ n, are guaranteed to be zero. Hence by (4.4 stability is guaranteed as expected [7]. However, if the flux function is non-linear, then the collocation projection employed to construct will introduce aliasing errors; that is to say the modal energies of (given by the first term on the right hand side of (5.4 will be different to the corresponding modal energies in f ˆ (given by the second term on the right hand side of (5.4. Such a phenomenon occurs because the collocation projection will in general under-sample f ˆ. Consequently high-frequency (under-resolved modes of will contribute (erroneously to the energies of lower-frequency resolved modes (for further details see, for example, the article of Kirby and Sherwin [9], or the textbooks of Karniadakis and Sherwin [10], and Hesthaven and Warburton [3]. As a result of these aliasing errors ˆɛ i will in general be non-zero, and thus in general the sign and magnitude of ˆɛ (and hence ɛ n cannot be guaranteed. Therefore by (4.4 stability of VCJH schemes can no longer be guaranteed if the flux function is non-linear. Such an instability is often referred to as an aliasing driven instability. There are various important points that should be noted about the aliasing driven instabilities that manifest when the flux function is non-linear: The instabilities are of the same form as those which afflict collocation based nodal DG schemes if the solution is under-resolved. If the solution (and hence f ˆ is well resolved, then aliasing errors, and hence aliasing driven instabilities, are effectively eliminated. The location of the solution points (at which the collocation projection is performed will have a significant impact on aliasing errors, and hence on aliasing driven instabilities. A sensible choice is to locate solution points at abscissa of the Gauss-egendre quadrature rule. To understand why, consider expanding (5.4as ˆɛ i = j j=0 l j i dr ˆ i dr. (5.5 Since l j is of order k and i is at most of order k 1, (5.5 can be written exactly as ˆɛ i = ˆ fj j=0 m=0 l j (ζ m i (ζ m ω m ˆ i dr (5.6 where ζ m and ω m are the abscissa and weights respectively of the Gauss-egendre quadrature rule. If it is now assumed that the solution points are located at the abscissa ζ m,then and hence ˆɛ i = f(ζ ˆ j δ jm i (ζ m ω m ˆ i dr (5.7 j=0 ˆɛ i = m=0 f(ζ ˆ j i (ζ j ω j j=0 ˆ i dr. (5.8 The summation in (5.8 can be recognized as the Gauss-egendre approximation of the integral term in (5.8. Such an approximation is of optimal accuracy (given a sampling

11 of the integrand at k + 1 points. Specifically, the approximation is exact for integrands up to order k + 1. The use of Gauss-egendre abscissa as solution points will therefore in general minimize the coefficients ˆɛ i, and thus minimize any aliasing errors. It can be noted that a similar argument follows for the Gauss-obatto-egendre abscissa. However, for such abscissa the approximation is only exact for integrands up to order k 1. Hence in general aliasing errors will be larger than if Gauss-egendre abscissa were employed. The fact that non-linear stability depends on solution point location is significant, since until now (based on linear analysis the stability of F schemes was considered to be independent of solution point location. In addition to minimizing aliasing errors, and hence aliasing driven instabilities, the solution points should also define a well conditioned basis set with which to represent the solution. In 1D (and hence via tensor product extensions in quadrilaterals and hexahedra Gauss-egendre and Gauss-obatto-egendre abscissa are suitable from this perspective (in fact Gauss-obatto-egendre abscissa can be viewed as optimal [3]. However, when selecting solution points in triangles, there is a conflict between the requirements of reduced aliasing and good conditioning. Finally, it can be noted that if the transformed discontinuous flux is obtained via an exact projection (as opposed to a collocation projection, such that is orthogonal to all polynomials of degree k, then according to (4.13 there will be no aliasing errors, since the spatial derivative of the approximate solution û δ is of degree k 1. Consequently, the resulting VCJH schemes will be non-linearly stable. However, it should be noted that performing such an projection exactly (or at least very accurately would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the F approach. 6 Conclusions It has been shown that VCJH schemes (at least in their standard form may be unstable if the flux function is non-linear. Such instability is due to aliasing errors, which manifest since F schemes (in their standard form utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. It has also been shown that the location of the solution points (at which the collocation projection is performed will have a significant effect on non-linear stability. This result is important, since linear analysis of F schemes implies that stability is independent of solution point location. Finally, it has been shown that if an exact projection is employed to construct an approximation of the flux, then aliasing errors will be eliminated, and non-linear stability will be recovered. However, performing such a projection exactly (or at least very accurately would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the F approach. It can be noted that in all above regards, non-linear stability properties of F schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of F schemes, which have hitherto been developed and analyzed solely in the context of a linear flux. Acknowledgements The authors would like to thank the National Science Foundation (grants and , the Air Force Office of Scientific esearch (grants FA and FA , the National Sciences and Engineering esearch Council of Canada and the Fonds de echerche sur la Nature et les Technologies du Québec for supporting this work.

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